A hidden order that emerges in the frustrated pyrochlore Tb 2+x Ti 2−x O 7+y with T c = 0.53 K is studied using specific heat, magnetization, and neutron scattering experiments on a high-quality single crystal. Semi-quantitative analyses based on a pseudospin-1/2 Hamiltonian for ionic non-Kramers magnetic doublets demonstrate that it is an ordered state of electric quadrupole moments. The elusive spin liquid state of the nominal Tb 2 Ti 2 O 7 is most likely a U(1) quantum spin-liquid state. PACS numbers: 75.40.Cx, 78.70.Nx, 75.10.Kt, 75.30.Ds Geometrically frustrated magnets have been actively investigated in condensed matter physics [1]. In particular, spin ice (SI), e.g. R 2 Ti 2 O 7 (R = Dy or Ho) [2,3], provides prototypical frustrated Ising magnets with the pyrochlore lattice structure [4], consisting of a three-dimensional network of cornersharing tetrahedra [ Fig. 1(b)]. It displays fascinating features such as a finite zero-point entropy [5] and thermally excited emergent magnetic or SI monopoles [6,7]. An intriguing theoretical proposal for a U(1) quantum spin liquid (QSL) state [8] has been made for variants of SI endowed with quantum spin fluctuations [9][10][11][12][13][14]. The U(1) QSL state [8-10] is characterized by an emergent U(1) gauge field producing gapless fictitious photons and by gapped bosonic spinon excitations carrying the SI magnetic monopole charge [8,9,13,15]. By increasing the transverse interaction, the system can undergo a phase transition from the U(1) QSL to a long range ordered (LRO) state of transverse spins or pseudospins representing electric-quadrupole moments for non-Kramers ions [9][10][11]. This state can be described as a Higgs phase [16][17][18][19][20].In a quest to QSL states in frustrated magnetic systems from both theoretical [21][22][23] and experimental [24,25] viewpoints, an Ising-like pyrochlore Tb 2 Ti 2 O 7 (TTO) is a potential candidate for a U(1) QSL: it has been reported to remain in a fluctuating spin state down to 50 mK without magnetic LRO [26,27]. However, the origin of this spin liquid state of TTO has been elusive for more than a decade despite many investigations (see Refs. [4,13,28] and references therein, and recent Refs. [29-31]), and is still under hot debate [13,28]. To solve this challenging problem of TTO, we start this investigation by postulating that the theoretically-proposed interaction between electric quadrupole moments of non-Kramers ions including Tb 3+ [the fourth term of Eq. interactions between magnetic dipole moments [the first three terms of Eq. (1)] and the perturbation through first excited crystal-field (CF) states [14,32], and by taking another assumption of Jahn-Teller (JT) distortion [28,33]. Under the present postulation, two ground states of off-stoichiometric Tb 2+x Ti 2−x O 7+y samples [34] will possibly be accounted for by the U(1) QSL (x < x c ) and electric quadrupolar (x > x c ) states of Ref. [9].In this Letter, we investigate the hidden order of Tb 2+x Ti 2−x O 7+y (x = 0.005 > x c ), because the electric qua...

The hidden ordered state of the frustrated pyrochlore oxide Tb2+xTi2−xO7+y is possibly one of the two electric multipolar, or quadrupolar, states of the effective pseudospin-1/2 Hamiltonian derived from crystal-field ground state doublets of non-Kramers Tb 3+ ions. These long-range orders are antiparallel or parallel alignments of transverse pseudospin components representing electric quadrupole moments, which cannot be observed as magnetic Bragg reflections by neutron scattering. However pseudospin waves of these states are composite waves of the magnetic-dipole and electric-quadrupole moments, and can be partly observed by inelastic magnetic neutron scattering. We calculate these spin-quadrupole waves using linear spin-wave theory and discuss previously observed low-energy magnetic excitation spectra of a polycrystalline sample with x = 0.005 (Tc = 0.5 K).

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Spin correlations of the frustrated pyrochlore oxide Tb2+xTi2−xO7+y have been investigated by using inelastic neutron scattering on single crystalline samples (x = −0.007, 0.000, and 0.003), which have the putative quantum-spin-liquid (QSL) or electric-quadrupolar ground states. Spin correlations, which are notably observed in nominally elastic scattering, show short-ranged correlations around L points [q = ( 1 2 , 1 2 , 1 2 )], tiny antiferromagnetic Bragg scattering at L and Γ points, and pinch-point type structures around Γ points. The short-ranged spin correlations were analyzed using a random phase approximation (RPA) assuming the paramagnetic state and two-spin interactions among Ising spins. These analyses have shown that the RPA scattering intensity well reproduces the experimental data using temperature and x dependent coupling constants of up to 10 th neighbor site pairs. This suggests that no symmetry breaking occurs in the QSL sample, and that a quantum treatment beyond the semi-classical RPA approach is required. Implications of the experimental data and the RPA analyses are discussed.

The ground states of the frustrated pyrochlore oxide Tb2+xTi2−xO7+y have been studied by inelastic neutron scattering experiments. Three single-crystal samples are investigated; one shows no phase transition (x = −0.007 < xc ∼ −0.0025), being a putative quantum spin-liquid (QSL), and the other two (x = 0.000, 0.003) show electric quadrupole ordering (QO) below Tc ∼ 0.5 K. The QSL sample shows continuum excitation spectra with an energy scale 0.1 meV as well as energyresolution-limited (nominally) elastic scattering. As x is increased, pseudospin wave of the QO state emerges from this continuum excitation, which agrees with that of powder samples and consequently verifies good x control for the present single crystal samples.